\Header{Wilcoxon}{Distribution of the Wilcoxon Rank Sum Statistic}
\alias{dwilcox}{Wilcoxon}
\alias{pwilcox}{Wilcoxon}
\alias{qwilcox}{Wilcoxon}
\alias{rwilcox}{Wilcoxon}
\keyword{distribution}{Wilcoxon}
\begin{Description}\relax
Density, distribution function, quantile function and random
generation for the distribution of the Wilcoxon rank sum statistic
obtained from samples with size \code{m} and \code{n}, respectively.\end{Description}
\begin{Usage}
\begin{verbatim}
dwilcox(x, m, n, log = FALSE)
pwilcox(q, m, n, lower.tail = TRUE, log.p = FALSE)
qwilcox(p, m, n, lower.tail = TRUE, log.p = FALSE)
rwilcox(nn, m, n)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, q}] vector of quantiles.
\item[\code{p}] vector of probabilities.
\item[\code{nn}] number of observations. If \code{length(nn) > 1}, the length
is taken to be the number required.
\item[\code{m, n}] numbers of observations in the first and second sample,
respectively.
\item[\code{log, log.p}] logical; if TRUE, probabilities p are given as log(p).
\item[\code{lower.tail}] logical; if TRUE (default), probabilities are
\eqn{P[X \le x]}{P[X <= x]}, otherwise, \eqn{P[X > x]}{P[X > x]}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
This distribution is obtained as follows.  Let \code{x} and \code{y}
be two random, independent samples of size \code{m} and \code{n}.
Then the Wilcoxon rank sum statistic is the number of all pairs
\code{(x[i], y[j])} for which \code{y[j]} is not greater than
\code{x[i]}.  This statistic takes values between \code{0} and
\code{m * n}, and its mean and variance are \code{m * n / 2} and
\code{m * n * (m + n + 1) / 12}, respectively.\end{Details}
\begin{Value}
\code{dwilcox} gives the density,
\code{pwilcox} gives the distribution function,
\code{qwilcox} gives the quantile function, and
\code{rwilcox} generates random deviates.\end{Value}
\begin{Note}\relax
S-PLUS uses a different (but equivalent) definition of the Wilcoxon
statistic.\end{Note}
\begin{Author}\relax
Kurt Hornik \email{hornik@ci.tuwien.ac.at}\end{Author}
\begin{SeeAlso}\relax
\code{\Link{dsignrank}} etc, for the \emph{one-sample} Wilcoxon
rank statistic.\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
x <- -1:(4*6 + 1)
fx <- dwilcox(x, 4, 6)
Fx <- pwilcox(x, 4, 6)

layout(rbind(1,2),width=1,heights=c(3,2))
plot(x, fx,type='h', col="violet",
     main= "Probabilities (density) of Wilcoxon-Statist.(n=6,m=4)")
plot(x, Fx,type="s", col="blue",
     main= "Distribution of Wilcoxon-Statist.(n=6,m=4)")
abline(h=0:1, col="gray20",lty=2)
layout(1)# set back

N <- 200
hist(U <- rwilcox(N, m=4,n=6), breaks=0:25 - 1/2, border="red", col="pink",
     sub = paste("N =",N))
mtext("N * f(x),  f() = true \"density\"", side=3, col="blue")
 lines(x, N*fx, type='h', col='blue', lwd=2)
points(x, N*fx, cex=2)

## Better is a Quantile-Quantile Plot
qqplot(U, qw <- qwilcox((1:N - 1/2)/N, m=4,n=6),
       main = paste("Q-Q-Plot of empirical and theoretical quantiles",
                     "Wilcoxon Statistic,  (m=4, n=6)",sep="\n"))
n <- as.numeric(names(print(tU <- table(U))))
text(n+.2, n+.5, labels=tU, col="red")
\end{ExampleCode}
\end{Examples}

